Final answer:
To find the distance between the points with polar coordinates (1, π/6) and (3, 3π/4), we can convert the polar coordinates to Cartesian coordinates and then use the distance formula. Converting (1, π/6) to Cartesian coordinates, we get (x, y) = (√3/2, 1/2). Converting (3, 3π/4) to Cartesian coordinates, we get (x, y) = (-3√2/2, 3√2/2). Using the distance formula, the distance between the two points is √(51 + 6√6)/4.
Step-by-step explanation:
To find the distance between the points with polar coordinates (1, π/6) and (3, 3π/4), we can convert the polar coordinates to Cartesian coordinates and then use the distance formula.
Converting (1, π/6) to Cartesian coordinates, we use the formulas x = r cos(θ) and y = r sin(θ). Plugging in the values, we get x = 1 * cos(π/6) = √3/2 and y = 1 * sin(π/6) = 1/2.
Converting (3, 3π/4) to Cartesian coordinates, we get x = 3 * cos(3π/4) = -3√2/2 and y = 3 * sin(3π/4) = 3√2/2.
Using the distance formula d = √((x₂ - x₁)² + (y₂ - y₁)²), we can find the distance between the two points.
d = √((-3√2/2 - √3/2)² + (3√2/2 - 1/2)²) = √((9/2 + 3√6/2 + 3/2) + (9/2 - 3√6/2 + 1/4)) = √(18/2 + 6√6/2 + 3/2 + 9/2 - 3√6/2 + 1/4) = √(45/4 + 3√6/2 + 3/2) = √(45/4 + 6√6/4 + 6/4) = √(45 + 6√6 + 6)/4 = √(51 + 6√6)/4.
So, the distance between the points with polar coordinates (1, π/6) and (3, 3π/4) is √(51 + 6√6)/4.